Subgroups — #21445417

Order 7808.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840158	C₂	1	2	yes	maximal
2	#16840142	C₂	1	1	yes	maximal, characteristic
3	#16840112	C₂	1	2	yes	maximal
4	C₈×#92418	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₄₄	—	2	2	no	—
7	#14519777	C₂²	1	2	yes	—
8	#14519708	—	2	4	no	—
9	#14519701	—	2	4	no	—
10	C₄×#92418	C₂²	1	1	yes	characteristic
11	C₄×#92418	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
13	#107×C₆₁	—	2	2	no	—
14	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₆₁	—	2	2	no	—
16	C₂×Dic₁₂₂	—	2	2	no	—
17	C₂×C₈×C₆₁	D₄	1	2	yes	—
18	C₄⋊C₄×C₆₁	—	2	2	no	—
19	C₄⋊C₄×C₆₁	—	2	2	no	—
20	C₄²×C₆₁	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₆₁	—	2	2	no	—
22	C₄⋊Dic₆₁	—	2	2	no	—
23	C₂×#92418	D₄	1	2	yes	—
24	C₂×#92418	D₄	1	2	yes	—
25	Dic₂₄₄	—	4	4	no	—
26	Q₈×C₆₁	—	4	4	no	—
27	C₂×C₄×C₆₁	—	4	2	no	—
28	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₆₁	—	4	2	no	—
32	Dic₁₂₂	—	4	4	no	—
33	C₈×C₆₁	SD₁₆	1	4	yes	—
34	#92418	—	2	2	no	—
35	#92418	Q₁₆	1	4	yes	complemented
36	C₂²×C₆₁	#127	1	1	yes	characteristic
37	C₄×C₆₁	—	8	2	no	—
38	C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
39	C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
40	C₄×C₆₁	#135	1	2	yes	—
41	Dic₆₁	—	8	2	no	—
42	#1707	—	61	1	no	maximal, Sylow
43	C₂×C₆₁	#492	1	1	yes	characteristic
44	C₂×C₆₁	#494	1	1	yes	characteristic
45	C₂×C₆₁	#495	1	1	yes	characteristic
46	#500	D₆₁	1	1	yes	characteristic
47	#498	—	61	2	no	—
48	#494	—	61	1	no	—
49	#487	—	61	2	no	—
50	C₈²	—	61	1	no	—
51	C₆₁	#1707	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₁₆	—	122	2	no	—
53	C₄⋊Q₈	—	61	2	no	—
54	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
55	#107	—	2	2	no	—
56	#106	—	122	4	no	—
57	#103	—	122	4	no	—
58	C₄×C₈	—	61	1	no	—
59	C₄×C₈	—	61	1	no	—
60	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	122	2	no	—
63	Q₁₆	—	244	4	no	—
64	C₂×C₈	—	61	2	no	—
65	C₂×C₈	—	61	2	no	—
66	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	122	2	no	—
70	C₄⋊C₄	—	122	2	no	—
71	C₄²	C₂²×D₆₁	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	244	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	244	2	no	—
76	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
77	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
79	C₈	—	61	4	no	—
80	C₈	—	122	2	no	—
81	C₈	#11758201	1	4	yes	—
82	C₂²	#14519773	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	488	2	no	—
85	C₄	C₂×#11758201	1	2	yes	—
86	C₄	Q₁₆×D₆₁	1	2	yes	—
87	C₄	#14519796	1	2	yes	—
88	C₂	#16840136	1	1	yes	characteristic
89	C₂	#16840140	1	1	yes	characteristic
90	C₂	#16840145	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14519773
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#500	C₂
socle	C₂²×C₆₁	#127
hypercenter	#500	D₆₁
hypocenter	C₆₁	#1707
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
35. #92418	Q₁₆	semidirect
51. C₆₁	#1707	semidirect